Achieving Absolute Convergence for the Dirichlet Eta Series via Compressive Re-Pairing (CORE): A Simple Tool for Exponentially Faster Computation of the Riemann Zeta Function

We introduce a compressive repairing (CORE) representation of the Dirichlet eta function, based on pairwise and higher-order difference blocks of the alternating series. The Dirichlet eta series provides a classical route to the analytic continuation of the Riemann zeta function to the half-plane Re(s) > 0. In this note we revisit η from a slightly different angle and study a decomposition into four real Dirichlet series (odd/even × cosine/sine) as well as a representation in terms of pairwise differences.  We analyse the real and imaginary parts on vertical lines s = σ + it, introduce a geometric interpretation of the contribution of each pair, and discuss higher-order difference blocks obtained by discrete differentiation of the basic Dirichlet series. From an analytic point of view, the pairwise representation furnishes an alternative analytic continuation of η into Re(s) > 0, together with quantitative bounds for the truncation error of the alternating series, and serves as a convenient working form of η on vertical lines. From a numerical point of view, higher-order difference blocks lead to improved asymptotic decay of the truncation error and hence to more efficient approximations for fixed target accuracy. We also sketch a simple model of the interplay between truncation error and floating-point rounding error and derive an approximate optimal truncation index N * , highlighting the role of difference-block representations as practical tools for high-precision computations of η(s) and ζ(s).